Internal Assessments
by bookwormbhavna
Summary: just my internal assessments from IB a high school program that mimics college courses ad in some cases grants students college credits for their scores. this includes my math, psychology history and french IA's as well as my submitted written tasks for english
1. Math

**IB Math Internal Assessment**

 _What is the relationship between the Fibonacci Sequence and the Golden Ratio and what are the real world applications of the two concepts?_

Bhavna L. Singh

Candidate Code: 001477-0174

Capuchino High School

IB Math HL 2

Session: May 2018

Word Count:2322

 **Introduction:**

The topic of research is the Fibonacci sequence and the golden ratio, specifically it's appearance in various subjects ranging from biology to art to architecture. The Fibonacci sequence is referenced as 'Fibonacci' because it is the common name of the Italian mathematician, Leonardo of Pisa, who introduced it to europeans in his work, _Liber Abaci,_ for the popularization of the Hindu-Arabic Number system in the 1202 AD. The sequence is referred to as 'Misrau Cha' , which in Sanskrit means two are mixed, in ancient Indian texts by Pingala 200 BCE reasserted by Virahanka in 700 CE, Halayudha 950 AD, Gopala 1135 AD, and about fifty years before Fibonacci introduced it Hemachandra but is hinted at in Hindu scripture from much earlier, so India is where the Fibonacci sequence is its origin despite its Italian name. The sequence itself starts from one, one is added to itself to get two and it continues with the term being added to the previous term to get the next, therefore bringing about the Fibonacci sequence. The sequence is often depicted in a spiral which can allow the the Fibonacci sequence to be applied to a physical structure. The golden ratio, also known as phi, is closely related to the Fibonacci sequence. It makes a major appearance in geometry which motivated Ancient Greek mathematicians and scholars to explore this mysterious concept which is evident because the first known written definition of the golden ratio was in Euclid's _Elements_ 300 BCE and in the features of Phidias' Parthenon built around 430 BCE. The golden ratio is calculated through division of a term with its previous term throughout the Fibonacci sequence as it oscillates towards the golden ratio, as demonstrated with 144/89 being 1.6179 and 233/144 is 1.61805 which we can see are extraordinarily close to the golden ratio 1.61804…, which therefore shows that as the limit of the Fibonacci sequence's division approaches infinity the division is equal to phi, making the range of application of the ratio the same as the sequence. This paper will explore the extent of these mathematical concepts appeal, because frankly the Fibonacci sequence and the golden ratio are incredibly fascinating to observe and apply, as well as them having immense informational depth.

 **Body:**

 **Fibonacci Sequence:**

 **Natural Structures**

There is so much to go for the application of the Fibonacci sequence as it shows up in various natural structures like in botany, I learned of the connection when I was looking at random Khan Academy videos when I was younger, and even in human biology, which i only learned when I explored more of it this year. For example in flowers the number of petals in a normal flower is most often a Fibonacci number like buttercups and wild roses having five petals each, corn marigolds have thirteen petals and some daisies have up to eighty nine petals. It's fascinating because these plant share only so many genes in common yet they still mimic the same pattern as one another. The way the petals are organized is incredible too since they are organized in Fibonacci spirals, which are just spirals but the number of those spirals is a Fibonacci number, like pine cones, for instance, have a few different types of spirals on them but the number of those individual spirals are a Fibonacci number. For example a pine cone could have thirteen positive steep spirals, thirteen negative steep spirals, twenty one positive gradual spirals and twenty one negative gradual spirals. Though some plants do not conform to this type of pattern, the fact that numerous plants do conform to this pattern is incredibly cool and definitely not randomly done. In biological structures it gets even more unusual. We can see the most simple patterns of singles or pairs with a pair of ears, eyes, hands and feet as well as a single nose, mouth, navel, or even more in fives with five fingers and five toes, but it's even seen in microstructures like DNA we can see that a full cycle of DNA is twenty one angstroms wide and thirty four angstroms long. This repeated pattern of Fibonacci numbers cannot be a coincidence with the vast difference between plants and humans to think that this pattern transcends the categories of living beings is incredible.

 **Man-made Structures:**

It's not just natural structures that mimic the Fibonacci sequence, man-made structures, like art , do as well. For instance, the Fibonacci spiral shows up in art throughout the world, the Fibonacci sequence can be depicted as a spiral that goes through squares with the dimensions of Fibonacci numbers, from least to greatest.

 _Wikipedia_

The curve is liberally used in art like in photography, most often photos have the subject in the center but the most appealing of the photos who do not have it center most often conform to the rule of thirds which utilizes the Fibonacci spiral for a more eye-catching photo. The parts where the curve is tighter is where the photo has the most detail and as the curve loosens there is less detail in the photo therefore carrying the eye along the curve. We can observe this quite clearly in the following photos, my friends in photography were quite surprised by the relationship between the Fibonacci sequence, a mathematical concept, and photography since they had not known of it before.

 _Pinterest_ _Jake Garn_

The way that it fits into photography is fascinating but it also shows up in classic paintings like _The Girl with Pearl Earring_ by Johannes Vermeer from the 17th century. We can see that similarly to photography most of the detail is in the tight curve of the spiral.

 _Johannes Vermeer_

Though the spiral is incredibly evident in visual art it is also prominent in music like on a piano eight notes make up a scale, thirteen notes make up an octave, there are eight white keys and five black keys. There are questions about whether or not Fibonacci numbers are related to pleasing musical frequencies but there is not enough definitive or convincing proof of this. The golden ratio has similar applications as the Fibonacci sequence, which will be explored next.

 **The Golden Ratio:**

The Golden Ratio also known as the golden mean or phi(?), as said before is related to the Fibonacci sequence through the continuous division of a Fibonacci number with its previous term,as the limit of that division approaches infinity it actually equals the golden ratio, as depicted below.

 _Story of Mathematics_

 **Natural Structures:**

The golden ratio can be applied to many natural structures and not just because of its relationship to the Fibonacci example, the optimal ratio between the consecutive parts of the finger is the golden ratio, and when the fingers are curled up they are the golden/Fibonacci spiral. It is not just the human hand but nearly everything about the human body can be related to the golden ratio, which is shown below.

 _The Golden Ratio in the Human body_

The image shows the ratio of the navel to the bottom of the feet and the navel to the top of the head as well as several other examples of the golden ratio, but does not include the fact that the ratio between the length of the humerus and the radius equals phi as well as several other examples. All of these measurements are based on bodies formed in optimal conditions soif the body is formed in non-optimal conditions the golden ratio does not always apply, since anything other than optimal conditions would disrupt the formation of the body making the golden ratio inapplicable to the areas affected. I tested this on the features of my classmates and myself to see if it was true and I saw that most often it is the facial features that are distorted due to non-optimal conditions because those are the fine details that would not matter as much in comparison to the length of bones or protecting our immune system. The golden ratio is not just in the human, it also shows up in botany just like the Fibonacci sequence. When plants are growing leaves they grow them in spirals and the angle between the leaves in the spiral is the golden angle. Phi on a circle would be 222.5 degrees and for peace of mind everyone uses the angle left over after subtracting 222.5 from 360, 137.5, which is called the golden angle. One of the reasons why plants form their leaves like this is so that the leaves would get the maximum amount of sunlight possible without blocking or being blocked by another leaf. This use of the golden angle leaves me astounded at this trait in not one species of plant but almost all species of plant that grew in optimal conditions, One can theorize that it is an evolutionary trait, which indicates to us that any plant that did not conform to the golden ratio did not survive as it would not get the nutrients needed for survival because of its non-golden-ratio conforming nature. This shows us the vast extent of the golden ratio in nature, from human biology to plants, but it does not just appear in nature since it also makes appearances in various man-made structures.

 **Man-made Structures:**

There are many man-made structures that exhibit the golden ratio, including art and architecture from around the world. The golden ratio is often included in art, mostly due it the proportion being aesthetically pleasing to the general public, as said earlier when discussing the Fibonacci spiral in art, there are many ways one can apply it to art from using the spiral to make the audience focus on a point of interest to making the dimensions of the object(s) in the art conform to the golden ratio. The golden ratio generally makes objects more appealing in any of its forms. Although the golden ratio has numerous applications in art it has great purpose in architecture. One of the oldest uses of the golden ratio is when Phidias used it in his construction, the Parthenon, in 430 BCE and was defined in Euclid's _Elements_ 300 BCE and was included in architectural theory by Luca Pacioli in his _Divina Proportione_ 1509 AD, a treatise made up of his three independent works involving the golden ratio, architecture, and geometry. The golden ratio is not only in Europe it also shows up in places from around the world. Places like the Taj Mahal 1684 AD use the golden ratio in the dimensions of the parts of the construction to have an aesthetic appeal rather than using the spiral to carry the eye. While some buildings like Toronto's CN tower still use it for aesthetic appeal with the total height of the tower compared to the height of the observation deck being equal to the golden ratio. Therefore showing the international extent of the golden ratio in man made structures.

 _The Taj Mahal_ _CN Tower in Toronto_

 **Conclusion:**

The golden ratio and the Fibonacci sequence are frankly fascinating as they have so many applications in the real world both mechanically and aesthetically. Not only that but there are so many avenues of research that are either unexplored or not explored deeply enough because there is just so much depth to the Fibonacci sequence and the golden ratio because they are not just rules and concepts to do with man made structure but they are inherent rules for all natural things and that is frankly amazing as it shows how everything is interconnected from little daisies to tall skyscrapers. It is not only man-made structures that make use of Fibonacci numbers and the golden ratio either. They used in religion as well with the thirteen days of funeral rites in Hinduism as well as the vast use of three throughout the world including Christianity's three aspects of god; the father, the son, and the holy ghost, the numerous European mythologies' Three Fates, the list can go on forever showing to us that the Fibonacci sequence and the golden ratio are not just scientific or mathematical, they play a part in religions all over the world making it a sociocultural aspect of this world giving it so much more depth. It is beautiful how both nature and humans all over the world have followed this code for millenia, which makes me so astounded and curious about this subject and researching this for months has made the subject so personal to me now.

 **Bibliography**

Dr R Knott: Fib ronknott DOT com. _Fibonacci Numbers and The Golden Section in Art, Architecture and Music_ , . . #modernarch.

"Fibonacci." _Wikipedia_ , Wikimedia Foundation, 18 Nov. 2017, wiki/Fibonacci#Fibonacci_sequence.

michaelsisk Follow. "Math 140 Fibonacci and golden ratio." _LinkedIn SlideShare_ , 14 July 2014, michaelsisk/math-140-Fibonacci-and-golden-ratio.

Marcus Frings, "The Golden Section in Architectural Theory", _Nexus Network Journal,_ vol. 4, no. 1 (Winter 2002), .

"Nature, The Golden Ratio, and Fibonacci too ..." _Nature, The Golden Ratio and Fibonacci Numbers_ , . .

"Phi in the human body." _Sacred Geometry_ , 16 Aug. 2012, /?q=en%2Fcontent%2Fphi-human-body.

Dharwadker, Sanjay. "Misrau Cha and Fibonacci." _Sanjay Dharwadker_ , 30 May 2015, /2015/05/30/misrau-cha-and-Fibonacci/.

Vihart. _YouTube_ , YouTube, 21 Dec. 2011, watch?v=ahXIMUkSXX0.


	2. History IA

Candidate Number : 01477-0174

To what extent was the Church involved in the resistance to fascist rule during World War 2 in Germany?

Bhavna Singh

Candidate Number: 001477-0174

Session: May 2018

International Baccalaureate Historical Investigation

Word Count:1976

 **Identification and Evaluation of Sources:**

The focus of this investigation is "To what extent was the Church involved in the resistance to fascist rule during World War 2 in Germany?", focussing on how the Church resisted and supported certain aspects of fascist rule under Hitler as well as how they were inactive in response to some aspects of the fascist regime. Christianity has been one of the most prominent religions throughout the years and so it is inevitable for there to have been a reaction from them in response to fascism. _The Catholic Church and the Holocaust 1930-1965_ by Michael Phayer is useful to the investigation as it focuses on the role of the Catholic Church in the Holocaust. _Christian Complicity? Changing Views on German Churches and the Holocaust_ by Robert P. Ericksen is also beneficial to the investigation as it explores how the Church supported Hitler and his regime.

Source: _The Catholic Church and the Holocaust, 1930-1965_ by Michael Phayer 2000

Published in 2000 by historian and Professor Emeritus at Marquette University Michael Phayer, the text's origin is valuable as the author has had decades of experience and time to research and write the book and the fact that he is the Ida E. King Distinguished Visiting Scholar of Holocaust Studies at Stockton University, adds credibility to his work. Moreso the publishing date of 2000 allows the historian hindsight on the events mentioned in the text.

However, there is the limitation that any church that supported the Holocaust might not have been forthcoming with that information in an effort to make themselves look better. Furthermore, the purpose of the book is valuable as it aims to inform people about the role of the Catholic Church during the Holocaust and is therefore important to my investigation as my investigation has to do with the role of the Church in resistance to Nazi rule in Germany.

Source: _Christian Complicity? Changing Views on German Churches and the Holocaust_ by Robert P. Ericksen 2009

The paper is sponsored by the United States Holocaust Memorial Museum, and although the museum clearly states that any assertions, opinions, and conclusions found in the paper are not reflected by the museum they still sponsored and published it, therefore adding credibility to the paper. The author is also a historian of the Holocaust working as a professor of history at Pacific Lutheran University indicating to us that the author has had years of study focussing on the Holocaust and can therefore be considered a credible source for information regarding the holocaust. In contrast there are limitations in regards to the purpose, since it aims to expose the complicity involving German churches and Nazis, people are less likely to believe it as people view religion as a peaceful aspect of humanity.

Word Count: 435

 **Investigation:**

"Love the Lord your God with all your heart and with all your soul and with all your mind and with all your strength.' The second is this: 'Love your neighbor as yourself.' There is no commandment greater than these." Mark 12:30-31 The verse has been interpreted to mean that loving 'God', yourself, and others is of the utmost importance, above all including the commandments that condemn others and is one of the more popular bible verses even to those not of the Christian religion. The verse is pertinent to the investigation as the the purpose of the investigation is to explore the extent to which the Church was involved in the resistance to fascist rule in Germany and having an understanding of the religion itself is important in trying to understand the Church's actions in the war and everyone is the Church's neighbor. So did they stay true to their commandments or did they go against what they hold so precious in the face of World War 2, a war that is made up Christian dominated countries ?

One way of interpreting the Church's actions in Germany is how they were supporting the Nazi's. "He[Gerhard Kittel] fully accepted the idea of a 'Jewish problem' even though Jews made up less than 1% of the German population". This Gerhard Kittel was an important figure in German Protestant theology and therefore held a great deal of influence in Germany. Kittel's cruelty towards the Jews only worsened after 1933. After that he spent a decade working in 'Walter Frank's Institute for the History of the New Germany', a Nazi facility for propaganda, and he was an important contributor to the 'Research Section on the Jewish Question' in the Institute. Not only that but the Nazi's set up the Reich Church which meant that there had to be Ayan representatives in the Church's hierarchy and they wanted to eliminate the Old Testament from what the Church studied. It is said that after thousands of years of hostility towards the Jews the Christians would be incapable to deny it and the contributions that Germany has made to Christianity, including Martin Luther and what he brought to advance the development of Protestantism. The Church also denied much of what was going on at the time "1942 the Holy See, or spokesmen for the pope in the office of secretary of state, said that rumors about mass murder could not be verified." Though later, when the Holocaust was evident to the rest of the world, Vatican personnel not the Pope said that "the crimes of one World War II combatant could not be condemned without condemning the crimes of another". The statement was released by Vatican personnel but not the Pope , which makes historians think about how the silence from the Pope and the ambiguity of Vatican personnel disconcerting to the majority of Christian devotees as they proceed to not reveal any of the reason for the Pope's silence. Consequently theological researcher Gregory Paul speaks of how the main reason that traditional Christian clergymen often cooperated with Hitler was because they were being bribed. He cites that 10% of taxes paid by German Catholics went towards the Church, which accrued the equivalence of $1 billion through the beginning of the Third Reich's relationship with the Church until their fall. Demonstrating how detrimental the Church was to those negatively affected by the war.

Another way to describe the Church's actions in the war is inactive, meaning that they may not have acted on certain aspects of Nazism at all. Information about mobile killing squads and death camps was collected piece by piece, in the Vatican and outside of it. At the time it could have only been labelled as a genocide if the information was all together By drawing accounts together from its diplomats stationed throughout Europe, the Holy See could have released the information and helped spread it but instead chose to slow down the spread of this knowledge as the Vatican had "no wish to give publicity to the issue." Thus showing that the Church had knowledge of such actions but chose not to act upon them. Though it can be argued that since they decelerated the spread of knowledge they were more active in supporting the Nazis than inactive towards the situations they were faced with. An example of withheld information includes the Holy See letting no one know that an Archbishop had dared to speak out against racism and genocide and whether it approved of his actions. This type of inaction was not just a singular instance it happened not just in Germany but in Croatia and Slovakia as well.

Finally on the point that the Church was very active in resistance to Nazi rule. For example, Dietrich Bonhoeffer, who ran the Confessing Church's preaching seminars in Zingst and Fürstenwal was a well known figure. Despite being banned from teach, writing anything for public consumption becoming forbidden for him as well as public speaking and seeing that the Confessing Church was weakened because of mass arrests, Bonhoeffer continued his resistance activity underground. Still making underground speeches to those who followed his sermons, Bonhoeffer was targeted by the German government and was eventually killed by the end of World War 2. Not only that during the German occupation of Poland, despite some of their anti-semitic values Christian Poles were horrified by the treatment of the Jews and were known to have helped them escape capture from the Nazi's. The irony of this astounds many in regards to the nature of those Poles as they preferred to prevent the suffering of their despised neighbors in contrast to encouraging it. Furthermore in regards to the Vatican one can say that when the Pope was dealing with the murdering of Jews in a diplomatic manner instead of a moral pronouncement such as an encyclical, a letter written by the pope that is sent to all th3e Bishops of the Catholic Church. Meaning that even when the Church was going against the Nazi's they were doing it is an underhanded manner possibly so as to not get caught going against Nazi ideology, but that would highly unlikely as the Church has immense power at an international level thus they are not likely to be intimidated by Germany. Therefore showing the Church's activity in Germany during World War 2.

The Church response to aspects of the Second World War can be seen as proactive in regards to the atrocities of the war but one has to take into account the inactivity towards other aspects and situations as well as detrimental to the resistance through their support of the Nazi regime. Despite the naive beliefs of the majority of Christians the Church did have a role in persecuting others during the Holocaust and even going so far as to deny it as it was happening, however there were aspects in which the Church resisted the Holocaust. Though the question is whether the Church's efforts to resist outweigh that of their efforts to help Nazi Germany and what would history have been like if the amount of activity varied from what it really was.

 **Reflection:**

As I conducted my investigation of the subject of the extent to which the Church resisted the Nazi regime I was challenged with many methods and challenges having to do with history research. The bias towards the Church made it hard to find evidence of inactivity and cooperation with the Nazi's as there have been many instances of the Church's denial towards the atrocities of World War 2 as shown in my investigation. Some of my sources involved accounts of the actions of the Church and its representatives that may have been unreliable as the people interviewed about those accounts would have made certain aspects of their account make themselves out to be better people. Though, the book _The Catholic Church and the Holocaust_ helped immensely as the sources associated with the book led me to more research that furthered my investigation. Furthermore in order to gain a basic understanding I consulted the most popular holy book associated with the religion, the Bible, as I had a limited understanding on the religion, being a person not of the Christian religion.

However the development of an actual research method for this topic was difficult since I was incapable of going to a college library for a time, I resorted to using college and library databases for sources having to do with my topic and accesses online versions of the books I needed for the investigation during that time. Also looking through college syllabi having to do with World War 2 was helpful in researching books that were pertinent to my subject did help me find some reliable sources for my investigation in their online forms, therefore allowing me to conduct the investigation.

Bibliography

Bray, Gerald. Bible Interpretations: Past and Present. Cambridge. James Clark and Co., 1990.

Ericksen, Robert P. Christian complicity?: changing views on German churches and the Holocaust.

Washington, D.C. (100 Raoul Wallenberg Place, SW, Washington, D.C. 20024-2126):

United States Holocaust Memorial Museum, Center for Advanced Holocaust Studies,

2009.

Jeremy Noakes, ed., Nazism, 1919-1945, Vol. 4: The German Home Front in World War II.

Exeter: University of Exeter Press, 1998, pp. 594-96.

Paul, Gregory S. "The Great Scandal: Christianity's Role in the Rise of the Nazi's." Free Inquiry.

Phayer, Michael. Catholic Church and the Holocaust, 1930-1965. Indiana U.P., 2001.

Phayer, Michael. Pius XII, the Holocaust, and the Cold War. Bloomington and Indianapolis: Indiana University Press., 2008

The Holy Bible: containing the Old and New Testaments translated out of the original tongues and

with the former translations diligently compared & revised. New York: American Bible

Society, 1986.


End file.
